For any \({\gamma }_{1}\in \Gamma _{m_{1}}\)  and \({\gamma }_{2}\in \Gamma _{m_{2}}\) , we define function \(g_{{\gamma }_{1},{\gamma }_{2}}\)  on \({{W}}\times {{Y}}\)  as
\(g_{{\gamma }_{1},{\gamma }_{2}}(w,y)=T(y)\left[{u({\gamma }_{2}(w))-u({\gamma }_{1}(w))}\right]-\left[{A({\gamma }_{2}(w))-A({\gamma }_{1}(w))}\right]\) 
and define \({{G}}(m_{1},m_{2})\)  the class of functions as
\({{G}}(m_{1},m_{2})=\left\lbrace g_{{\gamma }_{1}, {\gamma }_{2}}|\ {\gamma }_{1}\in \Gamma _{m_{1}}, {\gamma }_{2}\in \Gamma _{m_{2}}\right\rbrace .\) 
With the fact that \({{F}}(m_{1},m_{2})=\left\lbrace e^{g},\;g\in {{G}}(m_{1},m_{2})\right\rbrace \)  and the exponential function is monotone on \({\mathbb {R}}\) , by [1]}[Proposition 42-(ii)], it is enough to prove the conclusion holds for the class \({{G}}(m_{1},m_{2})\) .
Let \(K=|m_{1}\vee m_{2}|\)  be the number of segments given by the refined partition \(m_{1}\vee m_{2}\)  and \({\mathcal {I}}_{1},\ldots ,{\mathcal {I}}_{K}\)  the resulted segments on \({{W}}\) . For any \({\gamma }_{1}\in \Gamma _{m_{1}}\) , we can rewrite it as
\({\gamma }_{1}(w)=\sum _{k=1}^{K}a_{k}\mathbb {1}_{{\mathcal {I}}_{k}}(w),\mbox{\quad where\ }(a_{1},\ldots ,a_{K})\in I^{K}\) 
and any \({\gamma }_{2}\in \Gamma _{m_{2}}\) ,
\({\gamma }_{2}(w)=\sum _{k=1}^{K}b_{k}\mathbb {1}_{{\mathcal {I}}_{k}}(w),\mbox{\quad where\ }(b_{1},\ldots ,b_{K})\in I^{K}.\) 
As an immediate consequence, for any \(g_{{\gamma }_{1},{\gamma }_{2}}\in {{G}}(m_{1},m_{2})\) , it can be rewritten as
\(g_{{\gamma }_{1},{\gamma }_{2}}(w,y)=\sum _{k=1}^{K}\left[{u(b_{k})-u(a_{k})}\right]\mathbb {1}_{{\mathcal {I}}_{k}}(w)T(y)-\sum _{k=1}^{K}\left[{A(b_{k})-A(a_{k})}\right]\mathbb {1}_{{\mathcal {I}}_{k}}(w).\) 
Therefore, \({{G}}(m_{1},m_{2})\)  is contained in a \(2K\) -dimensional vector space spanned by \(\left\lbrace \mathbb {1}_{{\mathcal {I}}_{k}}(w), T(y)\mathbb {1}_{{\mathcal {I}}_{k}}(w),\;k=1,\ldots ,K\right\rbrace \) . By Lemma 2.6.15 of [2]}, we conclude \({{G}}(m_{1},m_{2})\)  is VC-subgraph on \({{X}}={{W}}\times {{Y}}\)  with dimension not larger than \(2K+1\) .
